Transcript 12 linear equations

LINEAR
EQUATIONS
MSJC ~ Menifee Valley Campus
Math Center Workshop Series
Janice Levasseur
Equations of the Line
• Write the equation of a line given the slope
and the y-intercept
• Write the equation of a line given the slope
and a point
• Write the equation of a line given two
points
Equations of the Line
Write the equation of a line given the slope
and the y-intercept: m and (0, b)
• Write the equation of a line given the slope
and a point
• Write the equation of a line given two
points
Ex: Find an equation of the line with
slope = 6 and y-int = (0, -3/2)
Recall: slope-intercept form of a linear equation
y = mx + b, where m and b are constants
Given the y-int = (0, -3/2)  b = - 3/2
Given the slope = 6  m = 6
Putting everything together we get the equation
of the line in slope-int form:
y = m
6 x + -b 3/2
y = 6x – 3/2
Ex: Find an equation of the line with
slope = 1.23 and y-int = (0, 0.63)
Recall: slope-intercept form of a linear equation
y = mx + b, where m and b are constants
Given the y-int = (0, 0.63)  b = 0.63
Given the slope = 1.23  m = 1.23
Putting everything together we get the equation
of the line in slope-int form:
y = 1.23
m x + 0.63
b
y = 1.23x + 0.63
Equations of the Line
• Write the equation of a line given the slope
and the y-intercept
Write the equation of a line given the slope
and a point: m and (x1, y1)
• Write the equation of a line given two
points
Ex: Find an equation of the line
with slope = -3 that contains the
point (4, 2)
Start with the slope-intercept form of a linear equation
y = mx + b
Slope = - 3 
y = - 3x + b
What is b, though?
Use the given point (4, 2)  x = 4 and y = 2
y = - 3x + b  2 = - 3(4) + b
2 = -12 + b
14 = b
put it together
we have m and b 
y = - 3 x + 14
Ex: Find an equation of the line
with slope = -0.25 that contains
the point (2, -6)
Start with the slope-intercept form of a linear equation
y = mx + b
Slope = -0.25  y = -0.25 x + b
What is b, though?
Use the given point (2, -6)  x = 2 and y = -6
y = -0.25 x + b  -6 = -0.25(2) + b
-6 = -0.5 + b
-5.5 = b
put it together
we have m and b 
y = -0.25x – 5.5
Equations of the Line
• Write the equation of a line given the slope
and the y-intercept
• Write the equation of a line given the slope
and a point
Write the equation of a line given two
points: (x1, y1) and (x2, y2)
Ex: Find an equation of the line
containing the points (-2, 1) and (3, 5)
Point 1
Point 2
First, find the slope of the line containing the points:
Slope = m = rise = y1 - y2 = 1 – (5)
run
x1 - x2
-2 – 3
4 4


5 5
Now we have m = 4/5 and two points. Pick one point
and proceed like in the last section.
We have m = 4/5, the point (-2, 1), and y = mx + b
Slope = 4/5  y = 4/5x + b
What is b, though?
Use the given point (-2, 1)  x = -2 and y = 1
y = 4/5x + b  1 = 4/5(-2) + b
1 = (-8/5) + b
13/5 = b
put it together 
we have m and b 
y = 4/5x + 13/5
Ex: Find an equation of the line
containing the points (-4, 5) and (-2, -3)
Point 1
Point 2
First, find the slope of the line containing the points:
8

-4 – (-2)  2
Slope = m = rise = y1 - y2 = 5 – (-3)
run
x1 - x2
= -4
Now we have m = -4 and two points. Pick one point
and proceed like in the last section.
We have m = -4, the point (-4, 5), and y = mx + b
Slope = -4 
y = -4x + b
What is b, though?
Use the given point (-4, 5)  x = -4 and y = 5
y = -4x + b 
5 = -4(-4) + b
5 = 16 + b
-11 = b
put it together 
we have m and b 
y = -4x – 11
Ex: Find an equation of the line
containing the points (0, 0) and (1, -5)
Point 1
Point 2
First, find the slope of the line containing the points:
Slope = m = rise = y1 - y2 = 0 – (-5)
run
x1 - x2
0 – (1)
5

1
= -5
Now we have m = -5 and two points. Pick one point
and proceed like in the last section.
We have m = -5, the point (0, 0), and y = mx + b
Slope = -5 
y = -5x + b
What is b, though?
Use the given point (0, 0)  x = 0 and y = 0
y = -5x + b 
0 = -5(0) + b
0=0+b
0=b
put it together 
we have m and b 
y = -5x + 0
y = -5x
Equations of the Line
• Write the equation of a line given the slope
and the y-intercept
• Write the equation of a line given the slope
and a point
• Write the equation of a line given two
points
Parallel & Perpendicular Lines
•
When we graph a pair of linear equations,
there are three possibilities:
1. the graphs intersect at exactly one point
2. the graphs do not intersect
3. the graphs intersect at infinitely many points
•
We will consider a special case of situation 1
and also situation 2.
Perpendicular Lines (Situation 1)
• Perpendicular lines intersect at a right
angle
• Notation:
› L1: y = m1x + b1
› L2: y = m2x + b2
› L1 ^ L2
Nonvertical perpendicular lines have slopes that
are the negative reciprocals of each other:
m1m2 = -1 ~ or ~ m1 = - 1/m2 ~ or ~ m2 = - 1/m1
If l1 is vertical (l1: x = a) and is perpendicular to l2,
then l2 is horizontal (l2: y = b) ~ and ~ vice versa
Ex: Determine whether or not the
graphs of the equations of the lines
are perpendicular:
l1: x + y = 8 and l2: x – y = - 1
First, determine the slopes of each line by
rewriting the equations in slope-intercept form:
l1: y = - 1x + 8 and l2: y = 1x + 1
m1 = -1 and m2 = 1
Since m1m2 = (-1)(1) = -1, the lines are perpendicular.
Ex: Determine whether or not the
graphs of the equations of the lines
are perpendicular:
l1: -2x + 3y = -21 and l2: 2y – 3x = 16
First, determine the slopes of each line by
rewriting the equations in slope-intercept form:
l1: y = (2/3)x - 7 and l2: y = (3/2)x + 8
m1 = 2/3 and m2 = 3/2
Since m1m2 = (2/3)(3/2) = 1 = -1
Therefore, the lines are not perpendicular!
Parallel Lines (Situation 2)
• Parallel lines do not intersect
• Notation:
› L1: y = m1x + b1
› L2: y = m2x + b2
› L1 || L2
Nonvertical parallel lines have the same slopes
but different y-intercepts:
m1 = m2 ~ and ~ b1 = b2
Horizontal Parallel Lines have equations
y = p and y = q
where p and q differ.
Vertical Parallel Lines have equations
x = p and x = q
where p and q differ.
Ex: Determine whether or not the
graphs of the equations of the lines
are parallel:
l1: 3x - y = -5 and l2: y – 3x = - 2
First, determine the slopes and intercepts of each
line by rewriting the equations in slope-intercept
form:
l1: y = 3x + 5 and l2: y = 3x - 2
m1 = 3 and m2 = 3
b1 = 5 and b2 = -2
Since m1 = m2 and b1 = b2 the lines are parallel.
Ex: Determine whether or not the
graphs of the equations of the lines
are parallel:
l1: 4x + y = 3 and l2: x + 4y = - 4
First, determine the slopes and intercepts of each
line by rewriting the equations in slope-intercept
form:
l1: y = -4x + 3 and l2: y = (-¼)x - 1
m1 = -4 and m2 = - ¼
Since m1 = m2 the lines are not parallel.